However as the propellent is used up, the emptier rocket will accelerate faster, and the acceleration increases. Hence there is a need to be able to calculate the final speed after some or all of the fuel has been burnt, allowing for this fact.

The rocket equation relates the before and after mass of the rocket, the exhaust velocity and the overall change in speed, when the thrust is applied in a straight line.

The rocket equation neglects any effects of gravity, wind resistance and assumes a constant exhaust velocity relative to the rocket. You'll note that the rocket equation doesn't depend on the rate that the fuel is burned.

Ve is closely related to the specific impulse (Isp). In fact you multiply specific impulse by the sea level acceleration due to gravity to get Ve.